Hydrodynamic Instability Centrifugal Instability: T Driving force:...

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Hydrodynamic Instability

Low ∆T: motionless, pure thermal conduction

Higher ∆T: steady convection roll

Even Higher ∆T: unsteady, turbulent

Τ+∆T

T

fluid

Rayleigh-Benard Instability:

1. Why turbulence occur?

q Driving force: buoyancy

q Damping force: viscous dissipation

stability of a heated plate


Low Ω: laminar, concentric streamlines

Higher Ω: steady convection roll

Even Higher Ω: unsteady, turbulent

Centrifugal Instability: Ω

q Driving force: centrifugal force

q Damping force: viscous dissipation


Low Re: steady, axisymmetric, parallel, parabolic Poiseuille flow

Re >2000: sporadic bursts of turbulence alternate with laminar

(intermittence)

High Re: fully turbulent

Reynolds’experiment: circular pipe flow

v Transition from laminar to turbulent flow need not be simply a function of the base flow considered, but also of the amplitude and type of perturbations.

vMultiple paths to transition through different sequences of laminar flow instabilities are extremely possible.

Hopf bifurcation

22222

22222

)()(2

)()(2

yxyyxyyxdtdy

yxxyxxyxdtdx

+−+σ+µ+=

+−+σ+−µ=

Use polar coordinates:

1

2 53

=θ

−σ+µ=

dtd

rrrdtdr

02at 0 :orbit periodic

)0,0(),( :point fixed

42 =−σ+µ=

=

cc rrdtdr

yx

dynamic system:

µ<

2

Hopf bifurcation

101.1for no 2 −=σ−

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3. Vorticity and Vortex Stretching

landing



Mixing layer between helium and nitrogen

U2/U1 = 0.38;

(L is the width of the picture)

411112 106 ;7 ×=µρ=ρρ LU

Tensor Notation

( ) +=ε≠≠±

=εotherwise 0

1 if 1 123kjiijk

=

=δotherwise 0 if 1 ji

ij

iω⇔ω r

j

j

j j

j

x

u

x

uu

∂

∂⇔

∂

∂⇔⋅∇ ∑

=

3

1

r

=δii = 3

jrisjsirmrsmij δδ−δδ=εε

332211

3

1

δ+δ+δ=δ∑=i

ii

4

Tensor Notation

j

kijk

j k j

kijk x

uxu

u∂∂

ε⇔∂∂

ε⇔×∇ ∑∑= =

3

1

3

1

r

kjijkj k

kjijk uuu ωε⇔ωε⇔ω× ∑∑= =

3

1

3

1

rr

( )j

ij

j j

ij x

ba

xb

aba∂∂

⇔∂∂

⇔∇⋅ ∑=

3

1

rrPhysical Significance

1. related to the rotational tensor

xr

δConsider the relative velocity of two nearby fluid particles separated by a distance :

ji

j

j

ij

i

j

j

i

j

iji xx

u

xu

xx

u

xu

xu

xu δ

∂

∂−

∂∂

+δ

∂

∂+

∂∂

=∂∂

δ=δ21

21

jijjiji xxSu δΩ+δ=δ

ij

ijS

Ω

= strain tensor (angular deformation; straining)

= rotational tensor

kijkijjkijki ωε−=ΩΩε−=ω 21 ;


2. Twice the average angular velocity

Consider an infinitesimal circle around a point in space:

( ) ∫∫∫∫ ⋅ωπ=⋅×∇π=AA

Adr

Adur

rrrr21

21

sdr ∫

⋅π r

sdur

rr21

average angular velocity =

As r →0, ( ) nn rr ω=πωπ= 212

21

The average angular velocity around an infinitesimal circle =

half the vorticity component in the direction of the area enclosed by the circle.

Physical Significance

3. Proportional to the angular momentum of fluid particles whose inertia tensor has spherical symmetry

∫∫∫δ

δ×δρ=δV

VduxArrr

angular momentum about its centroid :

Consider a fluid region centered at : Vδ xr

Physical Significance

lkmlkm ωε−=Ω 21

jijjiji xxSu δΩ+δ=δ

∫∫∫δ

δδρ=V

mjjm VdxxI = inertia tensor

( ) jijmmijjmkmijk IIISA ω−δ+ε=δ 21r

If spherical symmetry , ( )ijij II δ= ω=δω=δ rr

IAIA ii or

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3. Vorticity and Vortex Stretching Mathematical Description

Incompressible Navier-Stokes flows of Newtonian fluid with constant properties:

×∇ ( ) ugpuuutu rrrrrrr 2

21

21 1 ∇

ρµ

++∇ρ

−=ω×−⋅∇+∂∂

( ) ugpuutu

urrrrr

r

210

∇ρµ

++∇ρ

−=∇⋅+∂∂

=⋅∇

( ) ( ) ω∇ρµ

+×∇+∇×ρ

−∇=∇⋅ω−ω∇⋅+∂ω∂ rrrrrrr 21 gpuut

DtDω

r= Lagrangian time change rate of vorticity

Mathematical Description

ω∇ρµ

+×∇+∇×ρ

−∇=rr 21 gpRHS

pressure toque

zero if barotropic, i.e. ρ = ρ(p)

nonzero when the geometric centroid is not the mass centroid

body-force torque

zero if body force is irrotational or conservative, i.e. , where U is some scalar function, called potential function

Ug −∇=r

nonzero if rotational body force, e.g. Coriolis force

viscous diffusion

( ) ω∇ρµ

+×∇+∇×ρ

−∇=−ω∇⋅+∂ω∂ rrrrr 21 gput

( )urr ∇⋅ωvortex stretching

Consider an infinitesimal material line element :lrδ

)(tlr

δ )( dtt +δlr

( )uuDt

D rlrrlr

∇⋅δ=δ=δ = rate at which two initially infinitely

close-by fluid particles separate

barotropic fluids, conservative body force, inviscid:

( )uDtD rrr

∇⋅ω=ω


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powerfulness of turbulent stretching

∫∫∫ ∂∂

ω=ω

=ωA j

ij

A

i

Ai dAx

udA

DtD

dADtD

( ) 0=ω=ω∂∂

= ∫∫C

jjiA

jij

dsnudAux


vorticity/material tube:

( )uDtD rrr

∇⋅ω=ω

stretched (thinning) ð rotation speed up

compressed (flatten)ð rotation slow down

barotropic fluids, conservative body force:

( ) ω∇ν+∇⋅ω=ω rrrr

2uDtD

stretching + diffusion

nonlinear + linear

magnitude:

( )j

i

j

i

j

ii

jj

ijiii xxxxx

uDtD

∂ω∂

∂ω∂

ν−

∂ω∂

ω∂∂

ν+∂∂

ωω=ωω21

usually > 0, production (stretching) ⇒ finer, stronger structures

viscous diffusion

always < 0,

viscous dissipation

coarse, weaker structures

3. Vorticity and Vortex Stretching 3. Vorticity and Vortex Stretching

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Ref: Couder and Basdevant, JFM 173, 225-251 (1986)


vortex merging


The image at the right depicts two counter-rotating vortices as they interact with a rigid circular cylinder. Their initial translation is caused by their interaction and the upward directed jet produced in the gap between them. Once they collide with the cylinder, they produce an unsteady separation and another pair of vortices having a rotation which opposes the original pair of vortices. This image was generated by Dr. S. Subramaniam and is undoubtedly part of the unsteady separation work being conducted by Professor van Dommelen's group in FSU.

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barotropic fluids, conservative body force:

( ) ω∇ν+∇⋅ω=ω rrrr

2uDtD

( )j

i

j

i

j

ii

jj

ijiii xxxxx

uDtD

∂ω∂

∂ω∂

ν−

∂ω∂

ω∂∂

ν+∂∂

ωω=ωω21

•When a balance between the stretching and the diffusion/dissipation processes, the finest scale (Kolmogorov’s, η) is reached.

•When ν is very small (or Reynolds number is extremely large), diffusion won’t be significant until the gradients have become sufficiently large. That is, η decreases with decreasing ν.

•Will η → 0 as ν → 0?


Vortex stretching does not exist in two-dimensional flows.

( )( )),,(,0,0

0),,,(),,,(tyxu

tyxvtyxuuω=×∇=ω

= rrr

( ) 0=∂∂

ω=∇⋅ωzu

urrr

ð Turbulence is always three-dimensional

3. Vorticity and Vortex Stretchingexample

( )( ) vzyxvUu rrrr

+β+α−βα=+= ,,

nal)(irrotatio 0 and sible)(incompres 0 =×∇=⋅∇ UUrr

vurrr

×∇=×∇=ω

Townsend, Proc. Roy. Soc. A208, 534 (1951)

Consider only special cases when : Uvrr

>ω=ω yExpect when steady,

( ) 323333 ω∇ν+∂∂

ω=ω∇⋅+∂ω∂

zU

Ut

r

( ) ( ) ω∇ν+∇⋅ω=ω∇⋅+∂ω∂ rrrrrr 2UUt

23

2

33

dy

ddy

dy

ων+αω=

ωα−⇒

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( )( )αν−⋅=ω=ω⇒

ων+αω=

ωα−

223

23

2

33

exp)0()( yyy

dy

ddy

dy

model (1): ( )zyU αα−= ,,0r

i.e. vortex sheet structures of thickness

rate stretchingratediffusion

~2αν

Check:

( ) ( )0,0),( )(,0,0 13 yvvy =⇒ω=ωrr ( ) ( ) 0=ω∇⋅=∇⋅ω⇒ rrrr vv


model (2): ( )zyxU αα−α−= 2,,r

( ) ( )321 2,, αωαω−αω−=∇⋅ω Urr

Vortex stretching:

0,)( 2133 ≈ωω>>ω=ω r

Expect when steady,

( ) 323333 ω∇ν+∂∂

ω=ω∇⋅+∂ω∂

zU

Ut

r

ων+αω=

ωα−⇒

drd

rdrd

rdrd

r 333 2


model (2): ( )zyxU αα−α−= 2,,r

( )αν−⋅=ω=ω⇒

ων+αω=

ωα−

2233

33

3

exp)0()(

2

rrr

drd

rdrd

rdrd

r

i.e. vortex tube structures of radius

rate stretchingratediffusion

~2αν

Check:

( ) ( )0),(,0 )(,0,0 3 rvvr θ=⇒ω=ωrr ( ) ( ) 0=ω∇⋅=∇⋅ω⇒ rrrr vv

3. Vorticity and Vortex Stretchingexample

( )( ) vzyxvUu rrrr

+β+α−βα=+= ,,

nal)(irrotatio 0 and sible)(incompres 0 =×∇=⋅∇ UUrr

vurrr

×∇=×∇=ω

Consider only special cases when is two-dimensional, i.e..vr

( )( ) ),,(,0,0

0),,,(),,,( 21tyx

tyxvtyxvv

ω=ω

=rr

( ) ( )( ) ( )UvUu rrrrrrr ∇⋅ω=+∇⋅ω=∇⋅ωvortex stretching:

Buntine and Pullin, JFM 205, 263 (1989)

convection: ( ) ( ){ } ( )ω∇⋅≠ω∇⋅+=ω∇⋅ rrrrrrr UvUu

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3. Vorticity and Vortex StretchingexampleT.S. Lundgren, Phys. Fluids 25, 2193 (1982)

~ same with those used by Buntine and Pullin

~ analytical solutions

( ) )( ,0 ,2)( ztaUUrtaUU zr ==−== θr

~ tube core surrounded by spiral vortex sheets

Nonlinearity Generates Scales

( ) ugpuutu

urrrrr

r

210

∇ρµ

++∇ρ

−=∇⋅+∂∂

=⋅∇

mxBvkxAu

coscos

==

If at some instant time, the flow has

then,

=∂∂

=∂∂

xv

u

xu

u

mxBmkxA

kxAkkxA

sincos

sincos

⋅−

⋅− ( )

( ) ( )( )xkmxkmABm

kxkA

−++−=

−=

sinsin2

2sin2

2

Hydrodynamic Instability Centrifugal Instability: T Driving force:...

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